Maths for economists

Maths for economists Problem Set I

Due: 09/17/2015

Total is 20 points. Show your work.

1 Study and graph (4 points)

Let the function f defined by f (x ) = x 3 − 10x 2 + 5x − 7

a Where is f increasing/decreasing?

b Where is f convex/concave?

c What are f ’s critical points? Are they minimums, maximums or inflection points? Explain.

d Use the previous information to graph f .

2 Derivatives (6 points)

Compute the first derivatives of the expressions below:

a (sin(x 3))3

b e

1 x 2

c (ln (ln(x )))2

Compute the partial derivatives of the expressions below:

a ln(x1 + x2 + x1 x2)

b e

1 x1

+2x 22

c √ (x 31 + x 22 )2 + x1

1

3 Integrals (3 points)

Evaluate the following integrals:

a ∫ 2 1

2x 3 √

1 + x 4d x

b ∫ 2 1

2x 3p 2 + x 4

d x

c ∫ 2 1

x ln xd x

4 Turning points of functions with several variables (3 points)

Find the turning points of the following functions, if any:

a f (x , y, z ) = 5x + 7y − z + 6 b f (x , y, z ) = 3x y z − x y + 4z − z 3

5 Lagrange multiplier (4 points)

Use a Lagrange multiplier to find all the critical points of f on the given surface and discuss whether they are minima, maxima or neither. f (x , y, z ) = x 2 − y 2 on the surface x 2 + 2y 2 + 3z 2 = 1

2